Intertwining of simple characters in GL(n)

TL;DR

This paper establishes a precise criterion for when simple characters in GL(n) over a non-Archimedean local field intertwine, linking it to endo-equivalence, and demonstrates that all simple characters serve as G-types.

Contribution

It proves that intertwining of simple characters corresponds exactly to endo-equivalence and that all simple characters are G-types in GL(n).

Findings

Intertwining is equivalent to endo-equivalence for simple characters.

Every simple character in GL(n) is a G-type.

Provides a characterization of simple characters in the representation theory of GL(n).

Abstract

Let $F$ be a non-Archimedean local field and let $G$ be the general linear group $G = \text{\rm GL}_n(F)$. Let $\theta_1$, $\theta_2$ be simple characters in $G$. We show that $\theta_1$ intertwines with $\theta_2$ if and only if $\theta_1$ is endo-equivalent to $\theta_2$. We also show that any simple character in $G$ is a $G$-type.